Interpolation, extrapolation

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Interpolation

Interpolation is a type of estimation, a function that allow to estimate the value (x, y) between two known data points (x1, y1 ; x2, y2).

A common interpolation method used on ECU is the linear interpolation. The interpolation is more precise on high resolutions maps (multiples points)

Linear interpolation

Linear-inter.png

The formula :

where:

  • x1, y1 are the first known coordinates
  • x2, y2 are the second known coordinates
  • x is the point with want at which we do the interpolation
  • y is the interpolated value

Using linear interpolation on 3D maps:

  1. Interpolate the row above and below
  2. Interpolate the two interpolated values together


Here is a example, consider the following map:

Example map (load, engine speed)
Load /RPM 100 150
1000 10 20
2000 20 40

To get the value at load : 125 and engine speed : 1500, we will need to do a linear interpolation since it is not in the map.

The table would look like this after step 1:

Example map (load, engine speed)
Load /RPM 100 125 150
1000 10 15 20
2000 20 30 40

Now we interpolate 15 and 30 together to get the value at 1500 rpm (step 2):

Example map (load, engine speed)
Load /RPM 100 125 150
1000 10 15 20
1500 22.5
2000 20 30 40

At load : 125 and engine speed at 1500, we get the value 22.5 with linear interpolation

Extrapolation

ECU does not work with extrapolation. Extrapolation is a an estimation of values that are outside the scope of a map data. There are multiple proposed methods for extrapolation : linear, polynomial, conic, etc.

Extrapolation is not reliable and need to be used with care (if used).